Shannon extensions of regular local rings. Lefschetz properties for Gorenstein graded algebras associated to Apery Sets.

Let R be a regular local ring of dimension d > 1. Recently, several authors studied the rings obtained as infinite directed union of iterated local quadratic transforms of R. Here, in the first two chapters we present some results about the ideal theoretic structure and GCD property for such rings and we discuss the more general case of local monoidal transform of R. In the third chapter, we study the Weak Lefschetz property of two classes of standard graded Artinian Gorenstein algebras associated in a natural way to the Apery set of numerical semigroups. To this aim we also prove a general result about the transfer of Weak Lefschetz property from an Artinian Gorenstein algebra to its quotients modulo a colon ideal.